The Lagrangian of fractional classical mechanics $L_\alpha(\dot{q},q)$ is defined as

$$ L_\alpha(\dot{q}, q)=p\dot{q}-H_\alpha(p,q) $$

Where

$$ p=\frac{\partial L_\alpha(\dot{q}, q)}{\partial \dot{q}}, \\ H_\alpha(p, q)=D_\alpha |p|^\alpha + V(q), \qquad 1 < \alpha \le 2 $$

How to obtain the following using above definitions?

$$ L_\alpha(\dot{q}, q)=(\frac{1}{\alpha D_\alpha})^{\frac{1}{\alpha-1}} \frac{\alpha-1}{\alpha}|\dot{q}|^{\frac{\alpha}{\alpha-1}}, \qquad 1 < \alpha \le 2 $$

Source: Fractional Quantum Mechanics, Nick Laskin, World Scientific, 2018, page 260.